Quantum mechanics and quantum field theory. Algebraic and geometric approaches

Igor Frolov, Albert Schwarz

Published 2023-01-10Version 1

This is a non-standard exposition of main notions of quantum mechanics and quantum field theory that also includes some recent results. It is based on algebraic approach where the starting point is an associative algebra with involution and states are defined as positive linear functionals on this algebra and on geometric approach where the starting point is a set of states considered as a convex subset of linear space. The exposition does not depend on textbooks in quantum mechanics. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in geometric approach. Particles are defined as elementary excitations of ground state (and quasiparticles as elementary excitations of any translation invariant state). It follows from this definition that the notion of identical particles is very natural. The scattering of particles is analyzed in the framework of generalization of Haag-Ruelle theory. The conventional scattering matrix does not work for quasiparticles (and even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of inclusive scattering matrix, closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) anf inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. It is shown that generalized Green functions and inclusive scattering matrices appear also in the formalism of L-functionals that can be identified with positive functionals on Weyl or Clifford algebras. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of symbol of operator; these arguments can be applied also in geometric approach. This result can be used, in particular, to give a simple derivation of diagram technique for generalized Green functions. The notion of inclusive scattering matrix makes sense in geometric approach (but it seems that one cannot give a definition of conventional scattering matrix in this situation). The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of observables. This text is based on first ten lectures of the course taught by A. Schwarz in the Spring of 2022; see www.mathnet.ru for lectures (in Russian) and slides (in English). Keywords: Inclusive scattering matrix; generalized Green function, geometric approach